Time-Varying Energy Landscapes and Temperature paths: Dynamic Transition Rates in locally Ultrametric Complex Systems

Abstract

In this work, we study the dynamics of complex systems with time-dependent transition rates, focusing on $p$-adic analysis in modeling such systems. Starting from the master equation that governs the stochastic dynamics of a system with a large number of interacting components, we generalize it by $p$-adically parametrizing the metabasins to account for states that are organized in a fractal and hierarchical manner within the energy landscape. This leads to a not necessarily time homogeneous Markov process described by a time-dependent operator acting on an ultrametric space. We prove well-posedness of the initial value problem and analyze the stochastic nature of the master equation with time-dependent transition-operator. We demonstrate how ultrametricity simplifies the description of intra-metabasin dynamics without increasing computational complexity. We apply our theoretical framework to two scenarios: glass relaxation under rapid cooling and protein folding dynamics influenced by temperature variations. In the glass relaxation model, we observe anomalous relaxation behavior where the dynamics slow down during cooling, with lasting effects depending on how drastic the temperature drop is. In the protein folding model, we incorporate temperature-dependent transition rates to simulate folding and unfolding processes across the melting temperature. Our results capture a "whiplash" effect: from an unfolded state, the system folds and then returns to an unfolded state (which may differ from the initial one) in response to temperature changes. This study demonstrates the effectiveness of $p$-adic parametrization and ultrametric analysis in modeling complex systems with dynamic transition rate, providing analytical solutions that improve our understanding of relaxation processes in material and biological systems.

arXiv comment32 pages, 9 figures

References in corpus (11)

• Analytical Solution of the Off-Equilibrium Dynamics of a Long Range Spin-Glass Model
• Fractal free energy landscapes in structural glasses
• p-Adic Mathematical Physics
• Mode-Coupling Theory of the Glass Transition: A Primer
• p-Adic Models of Ultrametric Diffusion Constrained by Hierarchical Energy Landscapes
• Application of p-adic analysis to models of spontaneous breaking of the replica symmetry
• $p$-Adic Mathematical Physics: The First 30 Years
• Wavelet analysis as a p-adic spectral analysis
• p-Adic Modelling of the Genome and the Genetic Code
• On the two-steps relaxation of mean-field glasses: p-spin model
• Ultrametric Diffusion, Rugged Energy Landscapes and Transition Networks